Backbone colorings of graphs with bounded degree

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a spanning subgraphH of G (the backbone of G), a backbone coloring for G and H is a proper vertex k-coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2. The minimal k ∈ N for which such a coloring exists is called the backbone chromatic number of G. We show that for a graph G of maximum degree ∆ with the backbone graph being a d-degenerated subgraph of G, the backbone chromatic number is at most ∆+ d+ 1 and moreover, in the case when the backbone graph being a matching we prove that backbone chromatic number is at most ∆+ 1. We also present examples where these bounds are attained. Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H. We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G, then the backbone chromatic number is at most ∆(G)− √